The present invention relates generally to a method of and apparatus for analyzing a signal representing a physical parameter to obtain information from the signal and to extrapolate information contained in this signal and/or to ignore an interfering component in part of the signal. In particular, the present invention relates to the analysis of a signal representing either only a first portion of a skew distribution, or a first portion of a skew distribution and a second portion containing the interfering component. Particularly, the present invention provides for a method of determining the flow rate of a fluid using a signal analysis technique.
In order to measure flow rate of a fluid theoretically the simplest method is to add an indicator substance at a known concentration and constant rate into the flow and to measure its concentration downstream. Provided perfect mixing has occurred, the dilution factor gives a ratio of the two flows in the steady state. This approach has been used to measure cardiac output.
A more practical method, which has been employed to measure cardiac output, is to add the indicator as a bolus and to measure the concentration time profile of the passage of this bolus at a point downstream. Typically the concentration rises to a rounded peak and falls away more slowly as the remaining indicator is washed out. Such a curve may be considered as a large number of elements of sufficiently short time duration that for each element the flow and concentration are effectively constant. The concentration of the indicator multiplied by the flow and the time duration of each element gives the quantity of marker passed by that element. The sum of these quantities should therefore equal the total quantity of indicator added as the bolus. If the flow is constant for the duration of the curve, i.e. is the same for each of the elements, it may be calculated simply as the quantity of indicator added to the flow, e.g. by injection for cardiac output measurement, divided by the total area under the curve. The shape of the curve is unimportant providing that the total area can be found.
The bolus injection method of measuring cardiac output has been identified as suffering from the problem of the indicator starting to recirculate before the first pass of the indicator has been completed. Because of this recirculation some method is required to estimate the area of the primary curve, i.e. the curve which would have appeared had there been no recirculation, and so the shape of the curve becomes important. In a paper "Studies on the Circulation IV Further Analysis of the Injection Method, and of changes in Hemodynamics under Physiological and Pathological Conditions" by Hamilton et al, American Journal of Physiology 1932, Vol. 99, Pages 534-551, it has been proposed that the down-limb of the curve would, in the absence of recirculation, be monoexponential. The curve was plotted semilogarithmically so that the straight line of the down-limb could be extrapolated to a low value of the indicator concentration. The area which would have been inscribed by the curve representing the first pass of the indicator could therefore be measured. The theoretical basis for this technique has been challenged and a major problem for the use of such a technique clinically is that when the cardiac output is low the degree of encroachment of recirculation (the secondary curve) on the primary curve increases so that the extrapolation becomes unreliable.
The present inventors have developed a technique for analyzing signals which exhibit a skew distribution which avoids the need to measure these signals over the complete distribution or to utilize the part of the skew distribution where there is interference from an extraneous component, e.g. recirculation in the clinical indicator dilution curve technique. This technique can be applied to the analysis of any signals which exhibit a skew distribution and for which the integral of the distribution is required. This technique is however particularly applicable to the clinical indicator dilution technique in order to overcome the problem of recirculation. Such an indicator dilution technique can utilize lithium as the indicator and the invention is therefore particularly suited for the analysis of the indicator dilution curve provided from the cation detector disclosed in WO93/09427.
Although the possibility of modelling indicator dilution curves as a lognormal distribution which is a particular skew distribution has been disclosed in a paper entitled "An Empirical Formula for Indicator Dilution Curves as obtained in Human Beings" by Stow et al (Journal of Applied Physiology (1954), Vol. 7, Pages 161-167), the proposed use of the lognormal distribution to describe the primary curve merely represented the fitting of a mathematical relationship to the data with no theoretical support of the use of such a distribution to describe the indicator dilution curve. This paper further does not disclose how to obtain the area under the curve without being able to some how or other measure the primary curve without compensating for the secondary curve caused by recirculation.
The present inventors have therefore investigated the lognormal distribution in particular for indicator dilution curves to determine whether the lognormal distribution is a useful approximation.
A variable has a lognormal distribution if the logarithm of the variable is normally distributed (Skew Distributions. in Statistical Theory with Engineering Applications. 1952 by D. Hald, published by John Wiley & Sons, New York, London, Pages 159-187).
The equation of the lognormal distribution is ##EQU1##
where .mu. and .sigma. are the mean and standard deviation respectively of the normal distribution from which the logarithmic transformation was obtained. FIG. 1 shows lognormal curves varying from .sigma.=0.1 to 1.0, with the values of .sigma. indicated above each curve. Skewness is a monotonically increasing function of .sigma., so the curves become more skewed as .sigma. increases.
To determine whether the lognormal distribution is a good approximation for indicator dilution curves, the modelling of the effect of mixing an indicator of lithium into plasma at a concentration of 6 mM for one second was modelled passing through a cascade of six identical single pole filters each with a time constant of 1.3 seconds. The model used difference equations and the curves after each filter stage can be seen in FIG. 2. With three filter elements the curve is very close to a lognormal distribution and FIG. 3 illustrates an iterative best fit of a lognormal distribution to the third curve of FIG. 2 after the initial signal had passed through three filter elements. For the curve shown in FIG. 3 .mu.=1.76 and .sigma.=0.347.
It has also been determined by the inventors that over a range of skewness which is typically encountered in indicator dilution curves the lognormal distribution is a good approximation of the chi squared distribution. FIG. 4 illustrates the relationship between a chi squared distribution for 8 degrees of freedom (equivalent to four filter elements) and a lognormal curve where .sigma.=0.38 and .mu.=2.32. The curve represents the lognormal curve and the points represent points from the chi squared distribution. It can be seen from this that there is close approximation.
This work has provided theoretical justification for the use of the lognormal distribution in the analysis of indicator dilution curves and has led the inventors of the present application to develop an accurate and simple method of using the lognormal distribution to analyze indicator dilution curves.
Using the indicator dilution technique the cardiac output can be obtained from the equation ##EQU2##
where PCV is packed cell volume and may be calculated as haemoglobin concentration (g/dl)/33; this correction is needed because the lithium indicator is distributed in the plasma.
In order to calculate the cardiac output it is therefore necessary to determine the total area under the curve. One possible method is an iterative process to fit a lognormal curve to the data points. The use of the lognormal distribution to fit the indicator curve does however allow curves to be analyzed when encroachment of the secondary curve precludes the extrapolation technique used by Hamilton. The drawback of iteration is that it is relatively slow and occasionally unreliable.
The present inventors have therefore developed a technique which measures the area of a skew distribution (particularly the lognormal distribution) in order to overcome the disadvantages in the prior art. Although this technique is particularly applicable to indicator dilution curves in order to determine cardiac output, the technique is generally applicable to the analysis of any signal representing any physical parameter where the signal exhibits a skew distribution.